Oscillations, Waves, and Interactions - GWDG

74 D. Ronneberger et al.
Mechel found out that the sound attenuation in a channel the wall of which is
equipped with periodically spaced Helmholtz resonators is considerably reduced in
the presence of mean flow and may even be turned into sound amplification at certain
frequencies [2]. The relation between these frequencies, the flow velocity and
the spacing of the resonators revealed a close analogy between the observed sound
amplification and the amplification of electromagnetic waves in the so called travelling
wave tube which exploits the interaction between the electromagnetic wave
and an electron beam via a periodic structure [3]. However, the mechanism itself
by which mean-flow energy is converted to sound energy remained unclear. Besides
the sound amplification the excitation of loud tones was observed by Mechel when
the resonators were undamped. The latter phenomenon is known also from isolated
resonators the openings of which are exposed to grazing flow. A salient example of
such self-excited pressure oscillations has occurred in a gas transport system where
a pair of pipes with closed valves at the end branch off the main pipe in opposite
directions. The pressure amplitudes were so large that the flow velocity had to be
reduced in order to maintain safe operating conditions [4].
The phenomena observed by Mechel are two examples of aero-acoustic instabilities
which are based on the so called Kelvin-Helmholtz instability of vortex sheets. Such
vortex sheets form at the interface between the Helmholtz resonators and the interior
of the flow duct. While the sound amplification and the self-excited tones are closely
related to the inhomogeneity of the channel, namely to the spacing of the resonators
and maybe to the width of their openings, similar phenomena are also caused by
homogeneous compliant walls that bound internal and external flows.
The interest in the stability of fluid flow along compliant walls has been stimulated
by the experiments which M. O. Kramer had performed to explain Gray’s paradox:
Gray [5], wondering about the fast swimming speed of some dolphin species, had
calculated that the dolphin’s muscles had to deliver a multiple of the mechanical
power that is produced by the muscles of all other mammals unless the dolphins
are able to control the flow in the boundary layer around their body to remain
laminar. Kramer [6] speculated that the particular mechanical properties of the
dolphin’s skin stabilize the flow, and in fact, with special compliant coatings of his test
bodies he obtained an appreciable drag force reduction. However trials to reproduce
these spectacular results in other laboratories have failed. The seminal theoretical
investigation by Benjamin [7] and an impressive number of subsequent investigations
some of which are still in progress have nevertheless shown that the compliance of the
wall has indeed a strong effect on the stability of the flow boundary layer (see, e. g.,
Refs. [8,9]). The flow is destabilized in most cases because of a whole zoo of instability
modes that arise by the compliance of the wall, but under special circumstances a
stabilization can be achieved, and there is little doubt meanwhile that the onset of
turbulence can be delayed by appropriate coating of the wall.
In connection with sound propagation in acoustically treated flow ducts the existence
of the mentioned kind of instability has first been noted in theoretical studies
[10–16], however, there was no indication that these modes really exist in the
turbulent environment of practical flow situations. Nevertheless, the excitation of instability
modes at the leading edge of the compliant wall was taken into account, and
it was assumed that the growing instability modes loose their coherence by nonlinear